A119957 a(n) is the sum of p consecutive residues of 2^x modulo n, starting with a sufficiently large x and where p = period of binary representation of 1/n.
0, 0, 3, 0, 10, 6, 7, 0, 27, 20, 55, 12, 78, 14, 15, 0, 68, 54, 171, 40, 42, 110, 92, 24, 250, 156, 243, 28, 406, 30, 31, 0, 165, 136, 175, 108, 666, 342, 156, 80, 410, 84, 301, 220, 225, 184, 423, 48, 490, 500, 102, 312, 1378, 486, 440, 56, 513, 812, 1711, 60, 1830, 62
Offset: 1
Examples
a(1)=0 because 2^i mod 1 = {0,0,0,0,0,0,0,0,0...} and p=1;
a(2)=0 because 2^i mod 2 = {1,0,0,0,0,0,0,0,0...}, p=1, x>1;
a(14)=14 because 2^i mod 14 = {1,2,4,8,2,4,8,2,4,8,...}, p=3, x>1 ---> a=2+4+8=14;
a(35)=175 because 2^i mod 35 = {1,2,4,8,16,32,29,23,11,22,9,18,1,2,4,...}, p=12, x>0 ---> a = 1+2+4+8+16+32+29+23+11+22+9+18 = 175.
Crossrefs
Cf. A007733.
Formula
a(n) = Sum_{i=x..x+P-1} (2^i mod n) having: P=Period of binary representation of 1/n; x large enough for the period to start.
Comments